Abstract

We present a 3D lattice equation which is dual to the lattice AKP equation. Reductions of this equation include Rutishauser’s quotient-difference (QD) algorithm, the higher analogue of the discrete time Toda (HADT) equation and its corresponding quotient-quotient-difference (QQD) system, the discrete hungry Lotka-Volterra system, discrete hungry QD, as well as the hungry forms of HADT and QQD. We provide three conservation laws, we propose an N-soliton solution, and we conjecture that 2-periodic reductions to ordinary difference equations have the Laurent property.

1 A dual to the lattice AKP equation

In [6] a notion of a duality for ordinary difference equation was introduced. The idea is simple; given an OΔE, E=E(un,un+1,…,un+d)=0, with an integral, K[n]=K(un,un+1,…,un+d−1), the difference of the integral with its upshifted version factorises
K[n+1]−K[n]=EΛ. The quantity Λ is called an integrating factor. The equation Λ=0 is a dual equation to the equation E=0, both equations share the same integral. If E=0 has several integrals then a linear combination of them gives rise to a dual with parameters,

∑iaiKi[n+1]−∑iaiKi[n]=E(∑iaiΛi).

The notion extends to partial difference equations. Given a 2D lattice equation, E=E(uk,l,…,uk+d,l+e)=0, one now considers conservation laws:

P[k+1,l]−P[k,l]+Q[k,l+1]−Q[k,l]=EΛ.

Here the quantity Λ is called the characteristic of the conservation law. Again the equation Λ=0 or a linear combination,
∑iaiΛi=0, can be viewed as the dual equation to E=0.

In [4], six characteristics of conservation laws are given for a 3D lattice equation, namely the lattice BKP (Miwa [5]) equation

2 Conservation laws

As equation (2) arises as a dual to the lattice AKP equation, which has three parameters, this gives rise to three conservation laws.
Using shift operators σk, σl, σm and denoting the identity operation by I, we define

which is the third equation modulo (~σ−˙σ). In the same way one can obtain the first equation modulo (^σ−˙σ)
and the second equation modulo (^σ−1). In the sequel, we will refer
to the system of quotient and difference equations (3) and (4) as the Q3D-system.

We don’t have a formal proof, however, the result has been thoroughly checked, as follows: Taking particular values for a1,a2,a3 and a4, one can find rational points pi=(xi,yi,zi)∈Q3 such that Qi=0. Using N∈N points, one substitutes the N-soliton solution, which contains N arbitrary constants c1,…,cN, into the equation for fixed points (k,l,m)∈Z3. For example, taking (a1,a2,a3,a4)=(1,2,3,2) the following points

and we obtain similar expressions for the values of the 6-soliton solution
at the other 13 sites. Substituting these expressions into (10) gives 0.

5 Laurent property

Introducing the variable n=z1k+z2l+z3m, where we take z1,z2,z3 non-negative integers with greatest common divisor gcd(z1,z2,z3)=1, and performing a reduction τk,l,m→qn, one obtains
the ordinary difference equation

By introducing some special bi-orthogonal polynomials, in [1] the so-called discrete hungry quotient-difference (dhQD) algorithm and a system related to the QD-type discrete hungry Lotka-Volterra (QD-type dhLV) system have been derived, as well as hungry forms of the HADT-equation (hHADT) and the QQD scheme (hQQD). These systems are all reductions of the Q3D system,
or of the dual to the AKP equation, (2).